1

Cox Models for Vehicular Networks:SIR Performance and Equivalence

Jeya Pradha Jeyaraj, Student Member, IEEE, and Martin Haenggi, Fellow, IEEE

Abstract—We introduce a general framework for the modelingand analysis of vehicular networks by defining street systems asrandom 1D subsets of R2. The street system, in turn, specifiesthe random intensity measure of a Cox process of vehicles, i.e.,vehicles form independent 1D Poisson point processes on eachstreet. Models in this Coxian framework can characterize streetsof different lengths and orientations forming intersections or T-junctions. The lengths of the streets can be infinite or finite andmutually independent or dependent. We analyze the reliabilityof communication for different models, where reliability is theprobability that a vehicle at an intersection, a T-junction, ora general location can receive a message successfully from atransmitter at a certain distance. Further, we introduce a notionof equivalence between vehicular models, which means that arepresentative model can be used as a proxy for other modelsin terms of reliability. Specifically, we prove that the Poissonstick process-based vehicular network is equivalent to the Poissonline process-based and Poisson lilypond model-based vehicularnetworks, and their rotational variants.

Index Terms—Poisson line process, Poisson point process,stochastic geometry, and vehicular networks.

I. INTRODUCTION

A. Motivation and Related Work

Through vehicle-to-vehicle (V2V) communication, vehiclescan broadcast the information required for safety such astheir speed, brake status, position, etc., to other vehicles. Thiscan alert the vehicles about the events happening in theirvicinity that even the best sensors installed in the vehiclesmay fail to anticipate. Further, infrastructure nodes such asroadside units, smart traffic lights, etc., can inform the driversof the road/traffic conditions through vehicle-to-infrastructure(V2I) communication. High reliability is a key requirement forsafety-based V2V and V2I applications due to their mission-critical nature. Reliability or success probability is defined asthe probability that a broadcast message is received reliably ata certain distance. The factors that affect the reliability includethe locations of the transmitting and receiving vehicles andinterfering vehicles, and wireless medium.

An analysis of GPS traces of taxis in Beijing and Portoin [1] demonstrates that (i) the taxi locations cannot bemodeled as random points as in a 2D Poisson point process(PPP) neglecting the street geometry as claimed in [2], and (ii)it is desirable to model the intersections, which are critical forvehicle safety. Limiting the reliability analysis to a single or

The authors are with the Department of Electrical Engineering, Uni-versity of Notre Dame, Notre Dame, IN 46556, USA. (e-mail: {jjeyaraj,mhaenggi}@nd.edu).

This work was supported in part by Toyota Motor Corporation and by theU.S. National Science Foundation (grant CCF 2007498).

a small number of streets is certainly useful in its own right,albeit insufficient to obtain insights into the network behaviorat intersections and T-junctions [3]–[5] and in dense urbanscenarios [6]. Hence, street geometry is integral to vehicularnetwork modeling.

The street systems in different geographical regions maydiffer in their structure, street lengths, and degree of regularity.The authors in [7] have studied the street systems of 18 citiesin different parts of the world. They divide the street systemsinto two classes—(i) self-organized patterns that are formedhistorically without the control of any central agency, and (ii)planned regular grid-like patterns. Cities such as Ahmedabad(India), Cairo (Egypt), and Venice (Italy) are examples of self-organized patterns with unimodal street length distributions.Los Angeles, Richmond, and San Francisco in the UnitedStates are examples of grid-like patterns with multimodalstreet length distributions. It is noteworthy that even in thecities containing grid-like street patterns, the street lengths arefinite and vary significantly.

In the literature [8]–[14], street systems are most commonlymodeled as a random collection of lines with uniform orien-tations using Poisson line processes (PLPs). While assumingthat all streets are infinitely long may lead to useful results,it either overestimates the total interference or underestimatesthe local density of vehicles. For example, a street in a citythat is 2 km long may have a density of 50 cars per km.But extending it to an infinite street would mean such a highcar density extends to very remote rural regions far outsidethe city, which is not realistic. Further, in the PLP, each pairof streets forms an intersection and there are no T-junctions.Fig. 1 shows a part of Rome as an example, where few streetsare long enough to be approximated by infinite streets, andthere exist many T-junctions.

Consequently, we need street models that can characterizefinite variations in the street lengths, and intersections and T-junctions. The edges of Poisson-Voronoi tessellation, Poisson-Delaunay tessellation, and Poisson line tessellation are consid-ered to model streets of finite lengths in [15]. Estimators forthe probability densities of the inter-node distances are derivedfor these tessellations owing to their intractability. A simpleralternative is to use line segments or sticks as we show later. Inthis work, we introduce a general framework for the modelingand analysis of vehicular networks. Under this framework,we develop models that represent street systems varying fromregular grid-like patterns to irregular hodgepodges and charac-terize the uncertainty in the vehicle locations on the streets. Inparticular, the street lengths can be infinitely long or varyingfinitely and mutually independent or dependent resulting in

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Fig. 1: Part of Rome’s city center. Visual inspection shows a meanstreet length of about 250 m and many T-junctions.

intersections or T-junctions.An important question is whether we need a different

spatial model for each region. Alternatively, does there ex-ist an equivalence between the models such that a singlemodel is sufficient to analyze two different regions? If yes,a representative subset can be used to analyze a larger set ofmodels, reducing the computation time and costs associatedwith network design and planning. We show that some modelsdeveloped within the framework are equivalent, in a precisesense defined later. We use the mathematical toolsets fromstochastic geometry [16] to model and analyze random spatialpatterns.

B. Contributions

We present a general framework for the modeling andanalysis of vehicular networks, where the lengths of thestreets can be infinite or finite, mutually independent ordependent, and the street orientations can have different levelsof regularity. Lines or line segments (sticks) characterize thestreets. The street geometries of the spatial models that can becharacterized in this framework include but not limited to 1)the orthogonal grid with exponential spacing (OG) whose linesare of infinite length with orthogonal orientations, 2) the PLPwhose lines are of infinite length with irregular orientations, 3)the Poisson stick process (PSP) whose sticks can have randomlengths and orientations, and 4) the Poisson lilypond model(PLM) whose sticks grow in a random direction until theytouch another stick, mimicking the growth of lilies in a pond.

The OG, PLP, and PSP inherently form intersections,whereas in the PLM, the dependence between the sticks resultsin T-junctions. To the best of our knowledge, this is the firstwork to model and analyze vehicular networks with streetsystems formed by line segments involving intersections orT-junctions.

The vehicle locations on each street independently form 1DPPPs, i.e., the spatial models in the presented framework areCox vehicular networks. Our contributions are:• We introduce a street system partitioning approach that

decomposes a street system into points of different orders,where the order quantifies the number of street segments

covering the point. This decomposition permits a unifiedapproach to the performance analysis of vehicles atgeneral locations, intersections, and T-junctions.

• We derive the nearest-neighbor distance distribution inthe PSP-based vehicular network, and we show that thenearest-neighbor distance distribution in the PLM-basedvehicular network can be tightly approximated by that inthe PSP-based vehicular network.

• We derive analytical expressions for the probability thata vehicle at a general location, an intersection, or a T-junction successfully receives from its transmitter locatedat a certain distance, in each of the considered models.

• We introduce a notion of equivalence between the modelsbased on the success probability. The expression forsuccess probability of the typical vehicle in the PSP-basedvehicular network can be used to evaluate that in thevehicular networks formed by other models by mappingthe parameters such as street and vehicle intensities, andstreet length distribution. We show that the equivalencealso holds under random link distances.

• We show that the Cox vehicular networks behave likea 2D PPP in the low-reliability regime. In contrast, inthe high-reliability regime, the success probability of thetypical vehicle in the Cox vehicular networks tends to thatin the network formed only by the street(s) that the typicalvehicle belongs to. This corroborates earlier findings [1]that the vehicles cannot be generally modeled as 2D PPPsand that the street geometry plays an important role.

II. SYSTEM MODEL

Throughout the paper, we will use the definitions andnotations presented in this section unless otherwise stated. Letb(x, r) denote a disk of radius r centered at x, and o , (0, 0)denote the origin. Let | · |d denote the Lebesgue measure in ddimensions.

A. General Framework

Definition 1 (Street System). A street system S is a stationaryrandom closed subset of R2 with |S|2 = 0 that contains nosingletons or isolated points. Due to the stationarity,

E|S ∩ B|1 = τ |B|2 for Borel sets B ⊂ R2, (1)

where τ is the mean total street length per unit area.

|S|2 = 0 in Definition 1 implies that S is a random 1Dsubset of the plane consisting of lines, line segments or sticks,curved segments or arcs, that characterize the streets. Let Ξ ={ξ1, ξ2, . . .} be a collection of 1D subsets in R2 such that fori 6= j, |ξi ∩ ξj |1 = 0 and ξi ∪ ξj is not a 1D subset. The streetsystem S is the union of 1D subsets, i.e.,

S ,⋃ξ∈Ξ

ξ. (2)

The elements of S are points in R2 but those of Ξ are 1Dsubsets, which are sets themselves, and hence S 6= Ξ. Further,S uniquely characterizes Ξ and vice-versa, i.e., there existsa one-to-one correspondence between S and Ξ. Any streetsystem S can be partitioned as follows.

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Definition 2 (Street System Decomposition). Let Pm ,{z ∈ R2 : |S ∩ b(z, r)|1 ∼ mr, r → 0} denote the set of pointsof order m ∈ N in the street system S.

The sets Pm are disjoint, and their union equals S , i.e.,{Pm}m∈N is a partition of S. For m 6= 2, the sets Pm arecountable and form simple and stationary point processes. P2

is the only set with |P2|1 > 0, in fact, |P2|1 = ∞. Wehave P2 = S almost everywhere, i.e., P2 is an open set andS = cl(P2), where cl denotes the closure. P1 are endpoints,P3 are T-junctions, P4 are intersections, P5 are intersectionswith a T-junction, P6 are three-way intersections (three streetsintersecting at one point), etc. Let

M , {m ∈ N : P(Pm = ∅) = 0} (3)

denote the index set of the non-empty components. ThenS =

⋃m∈M Pm is called anM−indexed street system. Using

M, we can categorize different street systems. For example,a (2, 4)−street system refers to a street geometry withoutendpoints and T-junctions but with intersections.

Definition 3 (Vehicular Point Process). A vehicular pointprocess V ⊂ R2 is a Cox process with random intensitymeasure Υ(B) = λ|S ∩ B|1.

Equivalently, Υ(B) = λ|P2 ∩ B|1 since a.s. V ⊂ P2. Thisimplies that the vehicles form independent 1D PPPs on eachstreet. By Definition 1, the 2D intensity measure of V isE[Υ(B)] = λE[|S ∩ B|1] = λτ |B|2.

B. Vehicular Network Models

Here, we present a few models that fall under our frame-work. A street system may include curves or circles. In thiswork, we focus only on street systems formed by lines orsticks.

A line (an infinitely long street) L can be represented as

L(x, ϕ) = {(a, b) ∈ R2 : a cosϕ+ b sinϕ = x}, (4)

where (|x|, ϕ) are the polar coordinates of the foot of theperpendicular from the origin o to L.

Definition 4 (Line-based Poisson Street System). Consider amarked point process on R × Θ, whose ground process is aPPP Φ1 of intensity µ on R and marks are i.i.d. on Θ ⊆ [0, π).Let ν denote the distribution of Θ. The collection of linesΞL = {L(x, ϕ) : (x, ϕ) ∈ Φ1 ×Θ} with the intensity measureΛΞ(dxdϕ) = µdx× dν(ϕ) forms• an orthogonal grid with exponential spacing (OG) if

dν(ϕ) = 0.5δ0 + 0.5δπ/2,• a Poisson line process (PLP) if dν(ϕ) = dϕ/π.

The line-based Poisson street system is S =⋃L∈ΞL

L.

The OG and PLP inherently form intersections and henceare classified as (2, 4)−street systems. Streets of varyingfinite lengths can be represented using sticks. A stick S isdefined by its midpoint y, orientation ϕ, and half-length h,and represented as a closed set as

S(y, ϕ, h) = [y − hu(ϕ), y + hu(ϕ)], (5)

where u(ϕ) = (cosϕ, sinϕ). Alternatively, S(y, ϕ, h) = y +rotϕ([−h, h]), where rotϕ denotes the rotation by ϕ aroundo. Now, we are ready to define the stick-based street system.

Definition 5 (Stick-based Poisson Street System). Let Q ={(yi, ti)}, i ∈ N, denote an i.i.d. marked point process withthe ground process {yi} forming a 2D PPP Φ2 of intensityµ. Associate with yi a 2D mark ti = (ϕi, hi), where theorientation ϕi is i.i.d. on [0, π), and hi is the half-length.The countable collection of sticks ΞS = {S(yi, ϕi, hi)} forms• a Poisson stick process (PSP) if the half-lengths are i.i.d.

with some distribution FH .• a Poisson lilypond model (PLM)1 if each stick grows from

zero length at a constant rate on both sides until one of itsendpoints hits another stick, thereby forming a T-junction.

Then S =⋃S∈ΞS

S defines the stick-based Poisson streetsystem.

Let yi ≡ (γi, φi) in polar coordinates, where γi ∈ R+, andφi ∈ [0, 2π). The PSP has no T-junctions a.s., thus forming a(1, 2, 4)−street system, whereas the PLM has no intersectionsa.s. due to its touch-and-stop growth mechanism, thus forminga (1, 2, 3)−street system.

We only consider the street systems containing points oforder up to 4. The analyses shown in this work can be extendedto higher-order street systems as well.

We refer to the vehicular point processes formed by the OG,PLP, PSP, and PLM as the OG-PPP, PLP-PPP, PSP-PPP, andPLM-PPP, respectively. Fig. 2 depicts sample realizations.

C. Performance Metric and Types of Vehicles

Each vehicle broadcasts with probability p following theslotted ALOHA protocol. Then the intensity of active trans-mitters on each street in each time slot is λp. Our metric ofinterest is the success probability or reliability, which is theprobability of a vehicle successfully receiving the messagefrom the transmitter at a distance D. If a transmitter cancommunicate to a receiver at a distance D, then the otherreceivers within distance D are also highly likely to receivethe message. The transmitter can be another vehicle, a roadsideunit, a pedestrian, or any other node.

To define a meaningful network-wide metric, we focus onthe success probability of a representative vehicle (receiver)whose performance corresponds to the average of that of allvehicles. In point process theory, this representative vehicleis called ‘the typical point.’ In our context, it is ‘the typicalvehicle.’ As vehicles are located on the street(s), having avehicle at the origin implies that at least one street passesthrough the origin. Under expectation over the vehicular pointprocess, a vehicle conditioned to be at the origin becomesthe typical vehicle. Note that we can condition the typicalvehicle to be at any location since the vehicular network isstationary owing to the underlying stationary street system(Definition 1) and the homogeneity of the PPP. The typicalvehicle’s transmitter is assumed to be at a distance D fromthe origin.

1The PLM is denoted as LM-I in [17]. We note that the other model, LM-IIin [17], has similar properties as the PLM.

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(a) µ = 1 and λ = 0.1 (b) µ = 1 and λ = 0.1

(c) µ = 0.1 and λ = 0.1 (d) µ = 0.1 and λ = 0.1

Fig. 2: Snapshots of vehicular networks: (a) OG-PPP (b) PLP-PPP (c) PSP-PPP and (d) PLM-PPP. Lines or sticks denote the streets, and‘◦’ denote the vehicles. The stick lengths in (c) are Rayleigh distributed with parameter 2.

The performance of vehicular communication at intersec-tions and T-junctions is crucial as they are more prone toaccidents. In view of this, we evaluate the success probabilitiesof three kinds of vehicles: (i) the typical general vehicle whoseorder is 2; (ii) the typical intersection vehicle whose order is4; and (iii) the typical T-junction vehicle whose order is 3.The term typical vehicle refers to all the three types of typicalvehicles unless otherwise stated. Mathematically, the successprobability of the typical vehicle of order m at o is defined as

pm(θ) = P(SIR > θ | o ∈ Pm), m = 2, 3, 4, (6)

where SIR is the signal-to-interference ratio measured at o andθ parametrizes the target rate. The SIR for the typical vehiclewith its transmitter at distance D is given by

SIR =gD−α∑

z∈V gz‖z‖−αBz, (7)

where I =∑z∈V gz‖z‖−α is the total interference power at

the origin. The channel power gains g and gz are exponentiallydistributed with mean 1 (Rayleigh fading), and α is the path-loss exponent. (Bz)z∈V is an i.i.d. sequence of Bernoulli ran-dom variables with mean p, the probability that the transmitterz is active. Equipped with the performance metric, we nextdefine the equivalence of spatial models.

D. Equivalence

Definition 6 (Equivalence). Two spatial models A and B aresaid to be ε-equivalent with respect to the typical vehicle oforder m if the total variation distance of their SIR distributionsis at most ε, i.e., max

θ|pAm(θ)− pB

m(θ)| = ε, 0 ≤ ε ≤ 1. If thisholds with ε = 0, we call them strictly equivalent. Model A issaid to be asymptotically equivalent to model B in the lowerregime of θ if (1 − pA

m(θ))/(1 − pBm(θ)) → 1 as θ → 0, and

in the higher regime of θ if pAm(θ)/pB

m(θ)→ 1 as θ →∞.2

If the street system A has index set MA and the streetsystem B has index set MB, then the equivalence of A andB is defined only for the typical vehicles of orders MA ∩MB. If A and B are strictly equivalent, then either A or Bis sufficient to capture all the geographical regions that canbe characterized by them. Else, substituting one model for theother would depend on the complexity of the model and thevalue of ε, which we will discuss in detail in Section V-C.

2For all models, the success probability of the typical vehicle pm(θ)→ 1as θ → 0. Hence, considering ratios of success probabilities is not mean-ingful in this regime as they would all be equivalent. Similarly, the outageprobabilities, 1 − pm(θ), go to 1 as θ → ∞, so in this regime, it does notmake sense to consider the ratio of outage probabilities. To obtain non-trivialequivalence results, the quantities of interest need to go to zero.

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E. Further Notation

Conditioning on o ∈ P2 implies that a street passes throughthe origin, conditioning on o ∈ P3 implies that a street passesthrough the origin while another street ends at the origin,and conditioning on o ∈ P4 implies that two streets passthrough the origin. We denote by Vm = (V | o ∈ Pm) thepoint process of vehicles with the origin being on at least onestreet. Also, let Vmo denote the vehicles on the streets that passthrough or end at the origin. V! denotes the vehicles on the restof the streets, i.e., V! = Vm \Vmo . We index the streets basedon the distances of the perpendiculars from the streets to theorigin. Based on the indexing, Vk denotes the point processof vehicles on the kth street. Then Vmo = ∪1≤k≤dm/2eVk, andV! = ∪k>dm/2eVk. Let Imo and I! denote the interference fromthe vehicles in Vmo and V!, respectively. δ , 2/α, where α isthe path-loss exponent.

III. PROPERTIES OF VEHICULAR NETWORKS

Lemma 1. The mean total street length per unit area in theOG and PLP is τ = µ.

Proof: The total expected length of the lines that intersectb(o, r) is given by

E[|S ∩ b(o, r)|1] = µ

∫ π

0

∫ r

−r2√r2 − u2 dudv(ϕ)

= µ |b(o, r)|1. (8)

By Definition 1 and (8), we get τ = µ.

Lemma 2 ( [18], Eq. 9). The mean total stick length per unitarea in the PSP is τ = 2µE[H].

Let fH(h) denote the probability density function (PDF) ofthe half-lengths of the sticks.

Lemma 3. The half-lengths of the sticks that pass throughthe typical vehicle are distributed with density f̃H(h) =hfH(h)/E[H].

Lemma 3 is a case of the inspection paradox [19]. Thelength of the stick that passes through the typical vehicle isbiased by the fact that the mean number of points on the stickis proportional to its length. As a result, the half-lengths ofthose sticks follow the density function f̃H(h). For example,consider a case where half the streets are of length 10−2 andthe rest are of length 102, and the intensity of vehicles on eachstreet is 1. Then the typical vehicle lies on a long street witha probability of about 99.99%, which is very different fromthe 50% probability of the typical street to be a long one. Onthe other hand, in the PLM, a stick that ends at a T-junctionfollows the inherent distribution fH(h) since each stick hasexactly one such endpoint a.s. If all the sticks are of the samelength 2h0, then f̃H(h) = fH(h) = δ(h− h0).

In the lilypond model, the growth of a stick is determinedby the locations and the orientations of the other sticks sinceeach stick grows at a constant rate until one of its endpointshits another stick. This simultaneous touch-and-stop growthprocess makes it difficult, most likely impossible, to derivethe exact distribution of the half-lengths. However, a suitable

(a)

(b)

Fig. 3: (a) Fitting hf̂H(h)/E[H] to the half-lengths of the streets thatpass through the typical vehicle in the PLM. (b) Mean and varianceof number of neighbors to the typical general vehicle in the PLM-PPP vs. PSP-PPP with fH(h) = f̂H(h) = 2bh exp(−bh2). µ = 0.1and λ = 0.3. The value of b corresponding to µ = 0.01 is 0.0103.The intensity of 2D PPP is 2λµE[H], the 2D intensity of the PLM-PPP/PSP-PPP.

approximation for the distribution of the half-lengths can befound, as stated in Lemma 4.

Lemma 4. The half-lengths of the sticks in the PLM areapproximately Rayleigh distributed with PDF f̂H(h) =2bh exp(−bh2), h ≥ 0. It follows that E[H] ≈

√π/4b, where

b is proportional to the street intensity µ and can be estimatedfrom the empirical mean of the half-lengths.

Proof: See Appendix A.Combining Lemmas 3 and 4, we note that the PDF of half-

lengths of the sticks that pass through the typical vehicle inPLM can be approximated as f̃H(h) ≈ hf̂H(h)/E[H]. Fig. 3aillustrates their histogram and the fitted density functionsf̃H(h) and fH(h). We observe that hf̂H(h)/E[H] provides agood fit to the histogram, validating the approximation f̂H(h).

To appreciate the differences between the PSP-PPP andPLM-PPP, we consider the case where the half-lengths in thePSP are Rayleigh distributed as in the PLM. Let No(r) denote

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FPSP−PPPR (r) = 1−

[ ∞∫0

1

h

h∫0

exp(−λ`(γ, 0, 0))f̃H(h)dγdh

]m/2

× exp

(− µ

π

∞∫0

π∫0

2π∫0

r+h∫0

exp(−λ`(γ, φ, ϕ))γfH(h)dγdφdϕdh

). (9)

the number of neighbors to the typical vehicle at o within adistance r. Fig. 3b compares the mean and variance of No(r)in the PLM-PPP and PSP-PPP. The correlation among thestick lengths resulting from the touch-and-stop mechanism inthe PLM-PPP leads to a smaller variance of No(r) than inthe PSP-PPP, where the lengths of the sticks are independent.On the other hand, E[No] is the same for both the PSP-PPPand PLM-PPP when both follow the same distribution forhalf-lengths as their first-order statistics are the same. Also,the statistics of No for the PLM-PPP and PSP-PPP differsignificantly from that for a 2D PPP of equivalent intensity,highlighting the differences in having street geometry and not.

Lemma 5. The nearest-neighbor distance distribution for avehicular network with the street system characterized byDefinition 1 can be decomposed as

FR(r) = 1− (1− FR,Vmo (r))(1− FR,V!(r)), (10)

where FR,Vmo (r) is the probability of finding a neighbor in Vmowithin distance r, and FR,V!(r) is with respect to V!. FR,V!(r)also denotes the contact distance distribution, the distributionof the distance from an arbitrary location in the plane to thenearest vehicle in V .

Proof: See Appendix B.

Lemma 6. The nearest-neighbor distance distribution for theOG-PPP/PLP-PPP is FR(r) = 1 − exp(−λmr − 2µ

∫ r0

(1 −exp(−2λ

√r2 − u2 )du), where m ∈ {2, 4}.

Proof: See Appendix C.An alternative proof of the nearest-neighbor distance dis-

tribution for the case m = 2 in the PLP-PPP can be foundin [13]. Lemma 6 presents a slightly more general result thatalso applies to an intersection vehicle and the OG-PPP, whichcan be obtained by rotating each line L ∈ ΞL (Definition 4)constituting the PLP-PPP such that they are orthogonal to eachother.

Theorem 1. The nearest-neighbor distance distributionfor the PSP-PPP is given by (9), where `(γ, φ, ϕ) =`1(γ, φ, ϕ)1γ≤r + `2(γ, φ, ϕ)1γ>r. `1(γ, φ, ϕ), `2(γ, φ, ϕ) =|min(u1, h) ± min(u2, h)|, where u1, u2 = | − γ cos(φ −ϕ) ±

√r2 − γ2 sin2(φ− ϕ)|. f̃H(h) = hfH(h)/E[H] and

m ∈ {2, 4}.

Proof: See Appendix D.Fig. 4a shows the normalized mean distance E[Rn]/n

to the nth-nearest neighbor in the PLM-PPP and PSP-PPPwith Rayleigh distributed half-lengths. The rate of change inE[Rn]/n of the PLM-PPP from that of the PSP-PPP is at most

(a)

(b)

Fig. 4: Comparison of (a) mean distance from the typical generalvehicle to its nth-nearest neighbor and (b) nearest-neighbor distancedistributions in the PLM-PPP and PSP-PPP given by (9) withfH(h) = f̂H(h) = 2bh exp(−bh2). b = 1.04 for µ = 1, and 0.0103for µ = 0.01, and λ = 0.3.

6% for µ = 0.01, and 5% for µ = 1. Fig. 4b compares thenearest-neighbor distance distributions for different values ofµ. We see that the nearest-neighbor distance distribution inthe PLM-PPP is tightly upper bounded by that in the PSP-PPP in accordance with Fig. 4a, which shows that the meandistance to the nearest neighbor in the PLM-PPP is tightlylower bounded by that in the PSP-PPP. We presume that theinference obtained from Fig. 4b extends to the nth-nearestneighbor for n > 1 as well.

7

LPSP−PPPImo

(s) =

( ∞∫0

(1

2h

h∫−h

exp

(− λpsδ/2

(−w+h)s−δ/2∫(−w−h)s−δ/2

1

1 + v2/δdv

)dw

)f̃H(h)dh

)m/2. (11)

LPSP−PPPI!

(s) = exp

(− µ

π

∞∫0

π∫0

2π∫0

∞∫0

(1− exp

(− λp

h∫−h

(1 +

(F(γ, u, φ, ϕ)

sδ

)1/δ)−1

du

))γfH(h)dγdφdϕdh

). (12)

pPLM−PPP2 ≈ pPSP−PPP

2 = LPSP−PPPI2o

(θDα)LPSP−PPPI!

(θDα). (13)

pPLM−PPP3 ≈ pPSP−PPP

2 ×∞∫

0

exp

(− λpDθδ/2

2h

Dθδ/2∫0

1

1 + v2/δdv

)f̂H(h)dh. (14)

Conjecture 1. The distance from the typical general vehicle tothe nth-nearest neighbor in the PLM-PPP stochastically dom-inates that distance in the PSP-PPP with Rayleigh distributedhalf-lengths.

To facilitate the comparison of the success probabilities inthe general street-based Cox models and the homogeneousPPP, we recall the success probability of the typical vehiclein a PPP. Let Φd denote a stationary d−dimensional PPP ofintensity λd and cd denote the volume of a unit d−dimensionalball. In particular, c1 = 2 and c2 = π.

Lemma 7 ( [16], Sec. 5.2). The success probability ps of thetypical vehicle in Φd is

pΦds = exp(−cdλdDdθδΓ(1 + δ′)Γ(1− δ′)), (15)

where δ′ = d/α.

Next, we analyze the success probabilities of the typicalvehicle in Cox vehicular networks.

IV. SUCCESS PROBABILITIES

Using (7) and the notations in Section II-E, we express thesuccess probability pm of the typical vehicle of order m as

pm = P(g > θDαI)

= E[exp(−θDα(Imo + I!)]. (16)

We can simplify (16) for all the Cox vehicular networksconsidered but the PLM-PPP as

pm(a)= E[exp(−θDαImo )]E[exp(−θDαI!)] (17)(b)= LImo (s)LI!(s)|s=θDα , (18)

where (a) is due to the independence of the PPPs on thestreets, and (b) applies the definition of the Laplace transform.For the PLM-PPP, (17) does not hold as the length of eachstreet is dependent on that of other streets.

Proposition 1. The success probability of the typical vehiclein the OG-PPP/PLP-PPP is

pm = exp

(−mλpDθδ/2Γ(1 + δ/2)Γ(1− δ/2)

− 2µ

∫ ∞0

(1− LIx(θDα))dx

), (19)

where LIx(s) = exp(− λpsδ/2

∫∞vx

1

(1+v1/δ)√v−v0

dv)

with

vx = x2

sδ, and m ∈ {2, 4}.

Proof: See Appendix E.The success probability of the typical vehicle of order 2 in

the PLP-PPP is derived in [12]. The success probability (19)depends only on the distances of the interferers to the typicalvehicle, not their orientations. In Appendix E, we give a gen-eral proof that shows the effect of the order of the vehicle andthe irrelevance of the orientations on the success probability.

Proposition 2. The Laplace transform of the interferencefrom the vehicles on streets that pass through the typicalvehicle of order m ∈ {2, 4} in the PSP-PPP with half-lengthdensity function fH(h) is given by (11), where f̃H(h) =hfH(h)/E[H].

Proof: See Appendix F.

Proposition 3. The Laplace transform of the interferencefrom the vehicles on all but the streets that pass throughthe typical vehicle in the PSP-PPP with half-length densityfunction fH(h) is given by (12), where F(γ, u, φ, ϕ) = γ2 +u2 + 2γu cos(φ− ϕ).

Proof: See Appendix G.Following (18), the success probability of the typical vehicle

in the PSP-PPP is

pPSP−PPPm = LPSP−PPP

Imo(θDα)LPSP−PPP

I!(θDα), (20)

where LPSP−PPPImo

(s) and LPSP−PPPI!

(s) are given by (11) and(12), respectively.

Proposition 4. The success probabilities of the typical generalvehicle (order 2) and the typical T-junction vehicle (order3) in the PLM-PPP can be approximated as in (13) and(14), respectively. LPSP−PPP

I2o(s) and LPSP−PPP

I!(s) in (13) are

given by (11) and (12), respectively, with fH(h) = f̂H(h) =2bh exp(−bh2).

Proof: See Appendix H.Fig. 5 compares the success probabilities of the typical

general and intersection/T-junction vehicles. We omit the plotfor the success probability of the typical vehicle in the OG-PPP as it is the same as that in the PLP-PPP by Proposition 1.

8

We see that the success probability of the typical generalvehicle is higher than that of the typical intersection/T-junctionvehicle in all the Cox vehicular networks. The reason is thatas two streets pass through or end at the typical vehicle at anintersection or a T-junction, the received interference is highercompared to the typical general vehicle through which onlyone street passes.

Remark 1. Fig. 5c indicates that the success probability ofthe typical general vehicle in the PLM-PPP is tightly lowerbounded by that in the PSP-PPP with Rayleigh distributedhalf-lengths.

Remark 2. The approximation (14) to the success probabilityof the typical T-junction vehicle serves as a tight lower bound(see Fig. 5c). The integral term on the right-hand side of (14)is the Laplace transform of the interference from the vehicleson the street that has its endpoint at the origin. We canrephrase (14) as follows. The success probability of the typicalT-junction vehicle in the PLM-PPP is tightly lower bounded bythe success probability of the typical vehicle (with one streetpassing through) in the network formed by conditioning astreet in the PSP-PPP (with Rayleigh distributed half-lengths)such that one of its endpoints is at the origin.

Next, we analyze the equivalence between the spatial mod-els using their properties and the success probabilities of thetypical vehicle in those models.

V. EQUIVALENCE OF SPATIAL MODELS

To differentiate the street intensities of the models OG, PLP,PSP, and PLM, we denote them as µOG, µPLP, µPSP, andµPLM, respectively.

A. OG-PPP and PLP-PPP

Theorem 2. The PLP-PPP of street intensity µPLP is strictlyequivalent to the OG-PPP with street intensity µOG = µPLP.

Proof: The equivalence is a direct consequence of thesuccess probabilities of the typical vehicle in the OG-PPP andPLP-PPP given in Proposition 1.

B. PSP-PPP and PLP-PPP

Theorem 3. Let H , cH1, where c is a constant and H1

is a random variable with mean 1. The PLP is the limitingprocess of the PSP as c → ∞ and µPSP → 0 such that2µPSPE[H] = 2cµPSP = µPLP.

Proof: We formalize the heuristic arguments on therelation between the PLP and PSP given in [18]. FromDefinitions 4 and 5, we learn that the parameter spaces of lineand stick are different. To compare the line process with thestick process, we first establish compatible parametrizations.Let S′(r, φ, q) denote the infinitely extended stick S(y, ϕ, h),where q = yz is the distance between the midpoint of thestick y = (u, v) and z = (r cosφ, r sinφ), the closest pointto the origin from S′(r, φ, q). Fig. 6 illustrates S′(r, φ, q) andS(y, ϕ, h) using overlaid dashed and solid lines, respectively.

We can express y = (u, v) and ϕ in terms of (r, φ) and q asu = r cosφ− q sinφ, v = r sinφ+ q cosφ, and ϕ = φ− π/2.The differential element dudvdϕ equals drdφdq, i.e., thealternate parametrizations are equivalent. By (4), the line is justthe projection of the stick from the parameter space (r, φ, q) to(r, φ) when the latter is extended to infinity. However, we learnfrom Lemmas 1 and 2 that the properties of the PLP and PSPdiffer. Then, to obtain the PLP from PSP, we need to equatetheir statistical properties. Equating the mean total street lengthper unit area in the PLP and PSP given in Lemmas 1 and 2,we obtain 2µPSPE[H] = µPLP. For µPSPE[H] = cµPSP toremain finite, µPSP should go to zero as c→∞.

Corollary 1. The PSP-PPP, and its limiting case, the PLP-PPP, are strictly equivalent. As the PLP and PSP are identi-cally distributed as c→∞ and µPSP → 0, equivalence is notrestricted to PPPs on the streets but also holds for generalpoint processes of vehicles.

C. PLM-PPP and PSP-PPP

We learned from Proposition 4 that the success probabilityof the typical general vehicle in the PLM-PPP is approximatedby that in the PSP-PPP with the same half-length densityfunction as the PLM-PPP. Furthermore, from Figs. 4a and 4b,we infer that the nth-nearest-neighbor distance distribution inthe PLM-PPP is tightly upper bounded by that in the PSP-PPP. Consequently, we can deduce that the approximationto the success probability in the PLM-PPP is tight. Fig. 7compares the simulated success probability of the typicalgeneral vehicle in the PLM-PPP with its lower bound. Themaximum difference between the success probabilities in thePSP-PPP and PLM-PPP is ε = 0.0297 for µ = 0.01 and0.0219 for µ = 1. Though the PLM can characterize T-junctions, it is too complex to permit an exact analyticalexpression.

Remark 3. The PLM-PPP is ε−equivalent with ε � 1 tothe PSP-PPP with the same half-length distribution and streetintensity as the PLM-PPP. Consequently, the PSP-PPP servesas a good substitute for the PLM-PPP.

We see from Fig. 7 that the success probabilities of thetypical general vehicle in the PLM-PPP and PSP-PPP withRayleigh distributed half-lengths are even closer in the asymp-totic regions of θ than in the middle regions of θ. Theorem 4proves this observation formally.

Theorem 4. The PLM-PPP is asymptotically equivalent inboth the lower and upper regimes of θ to the PSP-PPP withthe same half-length density as the PLM-PPP.

Proof: First, we study the asymptotic behavior of thePSP-PPP as θ → 0 and ∞. Then, we compare them with thatof the PLM-PPP.

Lemma 8. As θ → 0, the PSP-PPP behaves like a vehicularpoint process formed only by the typical vehicle’s streets, i.e.,

1− pPSP−PPPm ∼ Θ(θδm/4). (21)

Proof: See Appendix I.

9

(a) (b) (c)

Fig. 5: Success probabilities of the typical general and intersection/T-junction vehicles in the (a) PLP-PPP (b) PSP-PPP and (c) PLM-PPP.λ = 0.3, D = 0.25, µ = 2, 0.1, and 0.3 for the PLP, PSP, and PLM, respectively. fPSP

H (h) = δ(h− 10). The equation numbers are givenin parentheses in the legends.

Fig. 6: Reparametrization of the PSP. The stick S(y, ϕ, h) ∈ ΞS

is extended to form a line S′(r, φ, q). The perpendicular from theextended stick is at a distance r from o and forms an angle φ withthe x−axis.

Fig. 7: Success probability of the typical general vehicle in the PLM-PPP vs. that in the PSP-PPP-based approximation given by (19) withfH(h) = 2bh exp(−bh2). b = 1.04 for µ = 1, and 0.0103 for 0.01.λp = 0.3 and D = 0.25.

Lemma 9. As θ → ∞, the success probability of the typicalvehicle in the PSP-PPP tends to that in a 2D PPP, i.e.,

pPSP−PPPm ∼ exp(−πλ2pD

2θδΓ(1 + δ)Γ(1− δ)), (22)

where λ2 = 2µλE[H] is the 2D intensity of the PSP-PPP.

Proof: See Appendix J.Equipped with the two lemmas, we continue with the proof

of the theorem. As θ → 0, for SIR > θ to hold, it suffices notto have any interferers within a small disk around the typicalvehicle. With high probability, the small disk intersects onlythe street(s) passing through the typical vehicle. Consequently,the system performance converges to that of only the typicalvehicle’s streets as θ → 0. The outage probability of the typicalvehicle due to its streets alone is proportional to λpθδm/4 asθ → 0. On the other hand, as θ → ∞, for SIR > θ, a largedisk around the typical vehicle must be devoid of interferers.It follows from Lemma 9 that the street geometry outside alarge disk does not matter as θ → ∞ since the PSP-PPP issimilar to a 2D PPP at a large scale.

Lemma 8 extends to the PLM-PPP as the interaction be-tween the sticks is irrelevant when θ → 0. Also, the successprobability of the typical general vehicle with respect to itsstreet alone in the PLM-PPP is the same as that in the PSP-PPP with the same half-length density as the PLM-PPP. InAppendix J, we reasoned that the PSP-PPP behaves like a2D PPP as θ → ∞ through mapping all the points on thesticks to their respective midpoints. The same logic holds forthe PLM-PPP as it is also formed by sticks whose midpointsform a PPP. As the PLM-PPP and PSP-PPP with the samehalf-length density as the PLM-PPP behave like 2D PPPs ofthe same intensities as θ →∞, they are equivalent as θ →∞.

Fig. 8 compares the success probabilities of the typicalgeneral vehicle in the street-based Cox vehicular networkswith that of the typical vehicle in 1D and 2D PPPs. Thesuccess probability of the typical vehicle in the PSP-PPP andPLM-PPP tends to that in the network formed only by thetypical vehicle’s street(s) as θ → 0 and to that of the 2D PPPas θ → ∞, as given in Theorem 4 and the lemmas therein.The same holds for the PLP-PPP, as established in [20, Th. 4

10

(a) (b) (c)

Fig. 8: Success probability of the typical general vehicle in the (a) PSP-PPP (b) PLM-PPP (c) PLP-PPP vs. that of the typical vehicle in1D and 2D PPPs. λ = 0.3, D = 0.25, µ = 0.1, 0.3, and 2 for the PSP, PLM, and PLP, respectively. fPSP

H (h) = δ(h − 10). The equationnumbers are given in parentheses in the legends.

and 5]. As vehicles on each street in the PLP-PPP form a 1DPPP, the PLP-PPP behaves like a 1D PPP as θ → 0.

Until now, we have assumed that the link distance D isfixed. Next, we discuss the equivalence between the spatialmodels when the link distances are random. Here, the successprobability is obtained by averaging the Laplace transform ofthe interference over the link distances.

D. Equivalence Under Random Link Distances

1) PLP-PPP vs. OG-PPP: The OG-PPP is just a rotationalvariant of the PLP-PPP. Both of them have the same statisticalproperties such as the mean total street length per unit area(Lemma 1), distribution of the distance to the nearest-neighbor(Lemma 6), and the Laplace transform of the interference(Proposition 1). Hence, the OG-PPP and PLP-PPP are strictlyequivalent even if the link distances are random.

2) LSP-PPP vs. PLP-PPP: The PLP is the limiting processof the PSP as the lengths of the sticks extend to infinity andstreet intensity tends to zero (Theorem 3). By the inherentnature of the PSP, it is equivalent to the PLP irrespective ofthe mode of communication.

3) PLM-PPP vs. PSP-PPP: We deduced from Figs. 4aand 4b that the nth-nearest-neighbor (or interferer) distancedistribution in the PLM-PPP is tightly upper bounded bythat in the PSP-PPP with Rayleigh distributed half-lengths.It follows that irrespective of the distribution of the distanceto the intended transmitter, the PLM-PPP and PSP-PPP withRayleigh distributed half-lengths are ε−equivalent. Fig. 9validates the above inference for the case where the typicalgeneral vehicle receives a message from its nearest neighbor(transmitter) through simulations.

Remark 4. The notion of equivalence enables us to consideronly a representative set of spatial models to obtain insightson the effect of the network parameters, thereby reducingthe computational costs and time associated with large-scaleexperiments and system-level simulations. The success proba-bilities of the typical vehicle in the OG-PPP, PLP-PPP, andPLM-PPP can be obtained from that in the PSP-PPP by asuitable mapping between the parameters as summarized inTable I.

Fig. 9: Success probabilities of the typical general vehicle thatreceives a message from its nearest neighbor. λp = 0.5, andfH(h) = f̂H(h) = 2bh exp(−bh2), where b = 1.04 for µ = 1,and 0.0103 for 0.01.

VI. CONCLUSIONS AND FUTURE WORK

We developed a Coxian framework for the modeling andanalysis of vehicular networks. The spatial models in thisframework can characterize different street geometries involv-ing intersections and T-junctions, and street lengths that canbe independent or dependent on each other. Streets of infinitelengths and different orientations forming intersections canbe characterized by PLPs and their rotational variants. PSPsand PLMs can model streets of varying finite lengths formingintersections and T-junctions, respectively. We evaluated thereliability of a vehicle-to-vehicle link in the Cox vehicularnetworks when the receiving vehicle is at an intersection, a T-junction, or a general location, and its transmitter is at a certainfixed distance. Our approach to defining the street systemas a union of points of different orders facilitates generalanalytical results for different types of typical vehicles. Theexpressions for the reliability can be used to investigate theinterplay among the network parameters such as data rate,street intensity, vehicle density, and the type of vehicle.

The concept of equivalence demonstrates that one does not

11

Table I: Conditions for equivalence between the spatial models.

Model A Model B Conditions for EquivalenceOG-PPP PLP-PPP µOG = µPLP

PLP-PPP PSP-PPP µPSP → 0 and c→∞ s.t. 2µPSPE[H] = 2cµPSP = µPLP

PSP-PPP PLM-PPP µPSP = µPLM, fPSPH (h) = fPLM

H (h)

need different spatial models to analyze the reliability of avehicle-to-vehicle link in different geographical regions. Themodels considered in the Coxian framework are equivalent interms of reliability. This implies that the expression for thereliability of the typical vehicle in the PSP-PPP is sufficientto evaluate that in the OG-PPP, PLP-PPP, and PLM-PPP byappropriately mapping the system parameters. Also, the vehic-ular networks behave like PPPs only in the asymptotic regimesof the reliability or data rate. Hence, the street geometry isrelevant to understand vehicular network behavior.

An interesting future extension would be to understand howtwo or more models developed in the Coxian framework canbe superimposed to represent an intricate geographic regionwith intersections and T-junctions, and streets and highways.Also, one may look for tractable models that characterizecurved streets, and examine whether they are equivalent tothe line/stick-based vehicular networks.

APPENDIX

A. Proof of Lemma 4

To prove Lemma 4, we make use of [17, Prop. 6.2], whichasserts the following: There exist a, b ≥ 0, such that thecomplementary cumulative distribution function (CCDF) of thehalf-lengths H in the lilypond model can be bounded as

1− FH(h) ≤ a exp(−bh2), h ≥ 0. (23)

Our goal is to find a tight approximation. By the properties ofthe cumulative distribution function, a ≥ 1. However, a > 1makes the bound (23) loose for small h, which implies thata = 1 is the natural choice. This reduces the bound (23) to theCCDF of the Rayleigh distribution. Fig. 10 fits the CCDF andPDF of the half-lengths to that of the Rayleigh distribution.We observe that the upper bound on the CCDF is loose forsmall values of h. Instead of choosing a value for b thatprovides an upper bound, we approximate the CCDF of thehalf-lengths with an appropriate value for b that minimizes thegap between the empirical and theoretical CCDFs. It followsfrom the Rayleigh approximation that the mean half-length isE[H] ≈

√π/4b. The value of b is found by equating E[H] to

the empirical mean of the half-lengths.Next, we study the relation between b and µ. Suppose we

scale the PLM by an arbitrary factor ν > 0. Then the streetintensity µ is scaled by 1/ν2 and the half-lengths of the sticksare scaled by ν, inversely proportional to

õ. As the PLM

retains its lilypond nature with scaling, the mean of the scaledhalf-lengths H ′ is scale-invariant, i.e.,

√π/4b′ ≈ E[H ′] =

E[νH] ≈√ν2π/4b. This implies that b′ = b/ν2, and thus b

scales with µ.

0 10 20 30 400

0.2

0.4

0.6

0.8

1

(a)

(b)

Fig. 10: Fitting the (a) CCDF and (b) PDF of the half-lengths to thatof the Rayleigh distribution. µ = 0.01.

B. Proof of Lemma 5

For the typical vehicle at the origin, the probability that itsnearest neighbor is at a distance greater than r, 1−FR(r), isgiven by

1− FR(r) = E[ ∏z∈Vm

1{z /∈ b(o, r)}]

(a)= E

[ ∏z∈Vmo

1{z /∈ b(o, r)}]E[ ∏z∈V!

1{z /∈ b(o, r)}]

= E[ dm/2e∏

k=1

∏z∈Vk

1{z /∈ b(o, r)}]

× E[ ∏k>dm/2e

∏z∈Vk

1{z /∈ b(o, r)}]

12

(b)= E

[ dm/2e∏k=1

E[ ∏z∈Vk

1{z /∈ b(o, r)}]]

︸ ︷︷ ︸1−FR,Vmo (r)

× E[ ∏k>dm/2e

E[ ∏z∈Vk

1{z /∈ b(o, r)}]]

︸ ︷︷ ︸1−FR,V! (r)

, (24)

where (a) and (b) exploit the independence of the 1D PPPs.The contact distance distribution GR(r) is the probability of

finding a vehicle within a distance r from an arbitrary locationin the plane, say the origin. No streets pass through or containthe origin a.s. This implies that GR(r) is defined with respectto V , which is equivalent in distribution to Vm \ Vmo = V!.Hence GR(r) = FR,V!(r).

C. Proof of Lemma 6

Using the probability generating functional (PGFL) of thePPP, we can express 1− FR,V!(r) in (24) as

1− FR,V!(r) = exp

(− 2µ

π∫0

r∫0

(1− exp(

− 2λ√r2 − u2))dudv(ϕ)

),

(a)= exp

(− 2µ

r∫0

(1− e−2λ√r2−u2

)du)

), (25)

where (a) follows from the independence between u and ϕ andthe fact that

∫R dv(ϕ) = 1 for the OG-PPP/PLP-PPP. Since

the vehicle locations follow a 1D PPP, with respect to thestreets that pass through the origin, we have 1− FR,Vmo (r) =(e−2λr)m/2.

D. Proof of Theorem 1

The probability that there are no vehicles on astick S(y, ϕ, h) within a distance r from the origin isexp(−λ|S(y, ϕ, h)∩b(o, r)|1). We derive |S(y, ϕ, h)∩b(o, r)|1as follows: The midpoint y is denoted as (γ, φ) in polarcoordinates. Using (5), we can express the points on a stickS(y, φ, h) of length 2h as (γ cosφ+u cosϕ, γ sinφ+u sinϕ),u ∈ (−h, h). Then the squared distance between a point onS(y, φ, h) and the origin is γ2 + u2 + 2γu cos(φ − ϕ). Thepoints of intersection of S(y, ϕ, h) on b(o, r) are at distances

u1, u2 = | − γ cos(φ− ϕ)±√r2 − γ2 sin2(φ− ϕ)| from the

midpoint of the stick. They follow from solving

γ2 + u2 + 2γu cos(φ− ϕ) = r2 (26)

with respect to u. For a stick S(y, ϕ, h) to intersect b(o, r), ymust be within b(o, r + h). Consider two cases—y ∈ b(o, r),and y ∈ b(r, r+h): 1. y ∈ b(o, r): Let |S(y, ϕ, h)∩b(o, r)| ,`1 for y ∈ b(o, r). When

a. S(y, ϕ, h) ∩ b(o, r) = S(y, ϕ, h): The stick lies withinb(o, r). Then `1(γ, φ, ϕ) = 2h.

Fig. 11: Realizations corresponding to cases (1a), (1b), (2a), and (2b).The midpoints of the streets are highlighted using orange filled ‘◦’.The points of intersection of the street on b(o, r) are at distances ui,i ∈ {1, 2}, from the midpoint of the street. ui in the cases (1b) and(2b) refers to u1 or u2.

b. S(y, ϕ, h) ∩ b(o, r) ⊂ S(y, ϕ, h): The stick is not fullycontained in b(o, r). For a stick that passes throughb(o, r), `1(γ, φ, ϕ) = u1 + u2. For a stick with only oneendpoint in b(o, r), `1(γ, φ, ϕ) = u1 + h or h+ u2.

Summarizing the above cases, we write `1(γ, φ, ϕ) =min(u1, h) + min(u2, h).2. y ∈ b(r, r + h): Let |S(y, ϕ, h) ∩ b(o, r)| , `2 for y ∈b(r, r + h). When

a. S(y, ϕ, h) ∩ b(o, r) = ∅: The stick does not intersectb(o, r). Then `2(γ, φ, ϕ) = 0.

b. S(y, ϕ, h) ∩ b(o, r) ⊂ S(y, ϕ, h): For a stick that passesthrough b(o, r), `2(γ, φ, ϕ) = |u1 − u2|. For a stick withonly one endpoint in b(o, r), `2(γ, φ, ϕ) = |u1 − h| or|h− u2|.

Therefore, `2(γ, φ, ϕ) = |min(u1, h)−min(u2, h)|. Then theprobability of not finding a vehicle on S(y, ϕ, h) within b(o, r)is exp(−λ`(γ, φ, ϕ)), where `(γ, φ, ϕ) = `1(γ, φ, ϕ)1γ≤r +`2(γ, φ, ϕ)1γ>r, 0 ≤ γ ≤ r + h. Fig. 11 illustrates the abovecases.

For the typical vehicle’s street to pass through or end atthe origin, its midpoint must belong to b(o, h). The midpointof the typical vehicle’s street of half-length h is at a distanceγ ∈ [0, h] from the origin. Using the above results and thedecomposition of the nearest-neighbor distance distribution inLemma 5, we can write 1− FR,Vmo (r) and 1− FR,V!(r) as in(27) and (28), respectively, by applying the PGFL of the PPP.

Note that (a) in (27) follows from the facts that (i) the rota-tion of the typical vehicle’s streets around the typical vehicledoes not change u1, u2, and hence we can set φ = ϕ = 0, and(ii) m/2 streets are independent. Substituting (27) and (28) in(10), we obtain (9).

E. Proof of Proposition 1

By (4), we can express the points on L(x, ϕ) as (x cosφ−u sinφ, x sinφ+u cosφ), u ∈ R. Let x refer to the kth-nearestpoint in Φ1 from the origin. Then Vk denotes the point processof vehicles on L(x, φ) (Section II-E). The Laplace transformof the interference LIx(s) from the vehicles on L(x, ϕ) to thetypical vehicle at the origin is given by

LIx(s) = E[ ∏z∈Vk

Egz[exp

(−sgz‖z‖−αBz

)] ]

13

1− FR,Vmo (r) = E[m/2∏k=1

E[ ∏z∈Vk

1{z /∈ b(o, r)}]]

(a)=

[ ∞∫0

1

h

h∫0

exp(−λ`(γ, 0, 0))f̃H(h)dγdh

]m/2. (27)

1− FR,V!(r) = exp

(− µ

π

∞∫0

π∫0

2π∫0

r+h∫0

exp(−λ`(γ, φ, ϕ))γfH(h)dγdφdϕdh

). (28)

= E[ ∏z∈Vk:Bz=1

1

1 + s‖z‖−α

](29)

(a)= exp

(− λp

∫ ∞−∞

1

1 +(x2+u2

sδ

)1/δ du

)(b)= exp

(− λpsδ/2

∫ ∞vx

1(1 + v1/δ

)√v − vx

dv

),

(30)

where δ = 2/α, (a) substitutes ‖z‖2 = ‖(x cosφ −u sinφ, x sinφ+u cosφ)‖2 and the PGFL of the PPP. λp is theintensity of the active transmitters for which Bz = 1, z ∈ Vk.(b) is due to the change of variable v = x2+u2

sδ, and vx = x2

sδ.

We learned from (18) that the success probability is theproduct of the Laplace transforms of the interference Imo andI!, which we derive below.

1) Laplace Transform of Imo : The Laplace transform of theinterference from V0 to the typical general vehicle (order 2)can be obtained by setting x = 0 in (30), i.e.,

LI2o (s) = exp(−2λpsδ/2Γ(1 + δ/2)Γ(1− δ/2)). (31)

For the typical intersection vehicle (order 4), as two streetspass through the origin, V4

o = V0∪V1 (Sec. II-E). Then LI4o (s)can be written as in (29) as follows:

LI4o (s) = E[ ∏z∈V0∪V1:Bz=1

1

1 + s‖z‖−α

](c)=

(E

[ ∏z∈V0:Bz=1

1

1 + s‖z‖−α

])2

, (32)

where (c) results from V0 being identically distributed as V1.It follows that LI4o (s) = L2

I2o(s).

2) Laplace Transform of I!: The aggregate Laplace trans-form of the interference from the vehicles on all the streetsthat do not pass through the typical vehicle is given by

LI!(s) = E[ ∏z∈V!

Egz[exp

(−sgz‖z‖−αBz

)] ](e)= E

[ ∏k>m/2

E[ ∏z∈Vk:Bz=1

1

1 + s‖z‖−α

]](33)

(f)= exp

(− µ

∫ π

0

∫ ∞−∞

(1− LIx(s)) dxdν(ϕ)

)(g)= exp

(− µ

∫ ∞−∞

(1− LIx(s)) dx

), (34)

where (e) follows from V! = ∪k>m/2Vk and the fact that theVk’s are independent 1D PPPs, (f) uses (30) and the PGFL of

the PPP with respect to x and ϕ, and (g) results from LIx(s)being independent of ϕ, and

∫R dν(ϕ) = 1. Substituting (31),

(32), and (34) in (18), we obtain (19).

F. Proof of Proposition 2

Using (5), we can denote the points on a stick S(y, φ, h) oflength 2h as (γ cosφ+u cosϕ, γ sinφ +u sinϕ), u ∈ (−h, h).The midpoint of the street that passes through the typicalvehicle is at a signed distance W ∼ U(−h, h) from theorigin. Then the endpoints of the street are at signed distances−W − h, and −W + h to the origin. As in (29), the Laplacetransform of the interference I2

o for the typical general vehiclecan be written as

LI2o (s) = E[ ∏z∈Vo:Bz=1

1

1 + s‖z‖−α

](a)= E

[exp

(− λp

−W+H∫−W−H

1

1 +(

usδ/2

)2/δ du

)](35)

(b)= E

[exp

(− λpsδ/2

(−W+H)

sδ/2∫(−W−H)

sδ/2

1

1 + v2/δdv

)], (36)

where (a) substitutes ‖z‖2 = ‖(u cosϕ, u sinϕ)‖2 and appliesthe PGFL of the PPP, and (b) results from the change ofvariable v = u

sδ/2. (c) evaluates the expectation with respect

to H using f̃H(h) rather than fH(h) based on Lemma 3. Forthe typical intersection vehicle, as two streets pass throughthe origin and the point processes on them are independent,the corresponding Laplace transform of the interference is thesquare of that given in (36) similar to (32).

G. Proof of Proposition 3

This proof uses the same notation as in Appendix F. LetA , R+×[0, 2π)×[0, π)×R+ and a = (γ, φ, ϕ, h) ∈ A. FromSection II-E, we have Vk, k > m/2, denoting the point processof vehicles on the kth street S(γ, φ, ϕ, h) that does not passthrough the origin. The Laplace transform of the interferenceLIa(s) from the vehicles on S(γ, φ, ϕ, h) is

LIa(s) = E[ ∏z∈Vk:Bz=1

1

1 + s‖z‖−α

](a)= exp

(− λp

h∫−h

1

1 + F(γ,u,φ,ϕ)1/δ

s

du

), (40)

14

LPSP−PPPI!

(θDα) ∼ 1− θµλp

π

∞∫0

π∫0

2π∫0

∞∫0

h∫−h

(D2

γ2 + u2 + 2γu cos(φ− ϕ)

)1/δ

γfH(h)dudγdφdϕdh = 1−Θ(θ). (37)

limθ→0

1− pPSP−PPPm = 1− lim

θ→0E[

exp

(− λpsδ/2

(−W+H)s−δ/2∫(−W−H)s−δ/2

1

1 + v2/δdv

)]m/2

(a)= 1− E

[1− λpsδ/2

(−W+H)s−δ/2∫(−W−H)s−δ/2

1

1 + v2/δdv

]m/2. (38)

1− pPSP−PPPm

(b)∼ 1− E

[1− λpsδ/2

2

(−W+H)2s−δ/2∫(−W−H)2s−δ/2

1

(1 + u1/δ)√u

du

]m/2(c)= Kαλps

δm/4. (39)

where (a) applies the PGFL of the PPP, ‖z‖2 = ‖(γ cosφ +u cosϕ, γ sinφ + u sinϕ)‖2, and F(γ, u, φ, ϕ) = γ2 + u2 +2γu cos(φ− ϕ).

Using (33), (40), and applying PGFL with respect to mid-points, orientations and half-lengths, we write the Laplacetransform of the interference from V! as

LI!(s) = E[ ∏k>m/2

E[ ∏z∈Vk:Bz=1

1

1 + s‖z‖−α

]]

= exp

(− µ

π

∫A

(1− LIa(s))γfH(h)dA), (41)

where dA = dγdφdϕdh.

H. Proof of Proposition 4

1) Success probability of the typical general vehicle in thePLM-PPP: We learned from Conjecture 1 that the probabilityof finding the nth-nearest neighbor closer is higher in thePSP-PPP than in the PLM-PPP. Consequently, the successprobability of the typical general vehicle in the PLM-PPP islower bounded by that in the PSP-PPP. As this inference isbased on a conjecture, we present the result for the successprobability of the typical general vehicle in the PLM-PPP asan approximation rather than a bound. By (16) and (18), wehave

pPLM−PPP2 = LPLM−PPP

I2o+I!(s)

≈ LPSP−PPPI2o

(s)LPSP−PPPI!

(s) |fH(h)=f̂H(h) .

(42)

2) Success probability of the typical T-junction vehicle inthe PLM-PPP: We have V3

o = V0 ∪ V1, where V0 denotethe vehicles on the street that passes through the origin, andV1 denote the vehicles on the street with one endpoint at theT-junction. The success probability of the typical T-junctionvehicle (order 3) is given by

p3 = E

[ ∏z∈V0∪V1:

Bz=1

1

1 + s‖z‖−α∏z∈V!:Bz=1

1

1 + s‖z‖−α

]

(a)≈ E

[ ∏z∈V0∪V!:Bz=1

1

1 + s‖z‖−α

]E

[ ∏z∈V1:Bz=1

1

1 + s‖z‖−α

]

(b)≈ LPSP−PPP

I2o(s)LPSP−PPP

I!(s)E

[ ∏z∈V1:Bz=1

1

1 + s‖z‖−α

]

(c)= LPSP−PPP

I2o(s)LPSP−PPP

I!(s)

×∞∫

0

exp

(− λpsδ/2

2hs−δ/2∫0

1

1 + v2/δdv

)f̂H(h)dh, (43)

where (a) assumes independence between V1 and V0∪V!, and(b) follows from (42). The integral expression in (c) can bederived similarly to (36) using the detail that for the streetwhose one endpoint is a T-junction at the origin, its otherendpoint is at a distance 2H from the origin. f̂H(h) is theapproximated density of the half-length of the street that endsat a T-junction.

I. Proof of Lemma 8

We can approximate LI!(θDα) (41) using Taylor’s series asθ → 0 as in (37). Using (35) and (37), as θ → 0, the outageprobability can be written as in (38), where (a) interchangesthe limit and expectation (by the monotone convergence the-orem), and applies Taylor’s theorem inside the expectation.We can further simplify (38) as θ → 0 as in (39), where (b)follows from the change of variable u = v2, and (c) resultsfrom the integral in (39) evaluating to a constant Kα as θ → 0.Note that Kα depends on α.

J. Proof of Lemma 9

Consider a model V ′ formed by mapping all the points oneach stick to its midpoint. The expected number of activetransmitters on each stick is 2λpE[H]. Since the midpointsfollow a 2D PPP of intensity µ, V ′ forms a non-simple 2DPPP with density 2µλpE[H]. It can be seen as a marked pointprocess whose ground process is a 2D PPP of intensity µ andthe marks 2λpE[H] refer to the multiplicity of the points. Let

15

M define the mapping from V to V ′. Using (6) and (7), thesuccess probability of the typical vehicle in the PSP-PPP canbe expressed as

pPSP−PPPm = P

(g > Dα

∑z∈V

gz‖θ−1/αz‖−αBz).

Similarly, the success probability of the typical vehicle in V ′is written as

pV′

m = P(g > Dα

∑z∈V

gz‖θ−1/αM(z)‖−αBz).

For sticks of half-length h, |‖z‖ − ‖M(z)‖| ≤ ‖z −M(z)‖ ≤h <∞. Then

|‖θ−1/αz‖ − ‖θ−1/αM(z)‖| → 0 as θ →∞. (44)

Then pPSP−PPPm → pV

′

m as θ →∞. The order m of the typicalvehicle is irrelevant as the mapping M does not distinguish anintersection from a non-intersection. The success probabilityof the typical vehicle pV

′

m can be obtained by substituting λ2 =2µλpE[H] in (15).

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Jeya Pradha Jeyaraj (Student Member, IEEE)received her M.Tech degree in Signal Processing,Communication, and Networks, from the Indian In-stitute of Technology, Kanpur, India, in 2013. Aftera stint as a Software Engineer at Cisco Systems,Pune, India, she is currently pursuing her Ph.D.degree in Electrical Engineering at the Universityof Notre Dame, Notre Dame, IN, USA, since 2016.Along with Ph.D., she is working towards her M.S.degree in Applied Computational and MathematicalStatistics. She held internship positions at InfoTech

Labs, Toyota Motor North America, Mountain View, CA, USA, in 2019, andat Wireless R & D, Qualcomm, USA, in 2020. Her research interests includevehicular networks, stochastic geometry, spectrum sharing, and data mining.

Martin Haenggi (S‘95-M‘99-SM‘04-F‘14) receivedthe Dipl.-Ing. (M.Sc.) and Dr.sc.techn. (Ph.D.) de-grees in electrical engineering from the Swiss Fed-eral Institute of Technology in Zurich (ETH) in1995 and 1999, respectively. Currently he is theFreimann Professor of Electrical Engineering and aConcurrent Professor of Applied and ComputationalMathematics and Statistics at the University of NotreDame, Indiana, USA. In 2007-2008, he was a visit-ing professor at the University of California at SanDiego, and in 2014-2015 he was an Invited Professor

at EPFL, Switzerland. He is a co-author of the monographs "Interference inLarge Wireless Networks" (NOW Publishers, 2009) and ‘Stochastic GeometryAnalysis of Cellular Networks’ (Cambridge University Press, 2018) andthe author of the textbook "Stochastic Geometry for Wireless Networks"(Cambridge, 2012), and he published 16 single-author journal articles. Hisscientific interests lie in networking and wireless communications, withan emphasis on cellular, amorphous, ad hoc (including D2D and M2M),cognitive, and vehicular networks. He served as an Associate Editor of theElsevier Journal of Ad Hoc Networks, the IEEE Transactions on MobileComputing (TMC), the ACM Transactions on Sensor Networks, as a GuestEditor for the IEEE Journal on Selected Areas in Communications, the IEEETransactions on Vehicular Technology, and the EURASIP Journal on WirelessCommunications and Networking, as a Steering Committee member of theTMC, and as the Chair of the Executive Editorial Committee of the IEEETransactions on Wireless Communications (TWC). From 2017 to 2018, hewas the Editor-in-Chief of the TWC. Currently he is an editor for MDPIInformation. For both his M.Sc. and Ph.D. theses, he was awarded the ETHmedal. He also received a CAREER award from the U.S. National ScienceFoundation in 2005 and three awards from the IEEE Communications Society,the 2010 Best Tutorial Paper award, the 2017 Stephen O. Rice Prize paperaward, and the 2017 Best Survey paper award, and he is a Clarivate AnalyticsHighly Cited Researcher.